Algebraic Geometry Motivated by Gauge Theory

撒迪厄斯9808529 这个项目的目的是一些问题，主要是在代数几何，但也在辛拓扑，这是预期有后果的规范理论，特别是塞伯格-威滕-弗洛尔理论。 首先，研究了黎曼曲面的对称乘积的量子上同调。 这个问题有很大的兴趣，在其本身的权利，是易于使用代数几何方法。 第二，考虑到有限阶映射的映射环面的Floer理论，导致了一个关于有限阶辛同胚的辛拓扑的独立有趣的猜想。 更长远的目标是正确理解与塞伯格-威滕理论相关的量子范畴。 代数几何和规范理论的接口，从圈群割线品种的其他几个主题，也将进行调查。 这是代数几何领域的研究。 代数几何是现代数学中最古老的部分之一，但在过去的四分之一个世纪里，它已经有了革命性的发展。在其起源，它处理的数字，可以定义在平面上的最简单的方程，即多项式。如今，该领域不仅使用代数方法，还使用分析和拓扑学方法，相反，在这些领域以及物理学、理论计算机科学和机器人技术中也得到了应用。

Thaddeus 9808529 This project is aimed at a number of problems, mostly in algebraic geometry but also in symplectic topology, which are expected to have consequences for gauge theory, and specifically for Seiberg-Witten-Floer theory. First, it is proposed to study the quantum cohomology of symmetric products of Riemann surfaces. This problem has great interest in its own right and is tractable using algebro-geometric methods. Second, considering the Floer theory for a mapping torus of a finite-order map leads to a conjecture, of independent interest, about the symplectic topology of a symplectomorphism of finite order. A longer-range goal is a proper understanding of the quantum category associated to Seiberg-Witten theory. Several other topics at the interface of algebraic geometry and gauge theory, ranging from loop groups to secant varieties, will also be investigated. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.